Образы линейных условий на плоскости с прямоугольной метрикой

Sylvester [7] proposed the following question in 1893. If a finite set of points in a plane is such that on the line determined by any two points of the set there is always a third point of the set, is the set collinear? Equivalently, given a finite planar set of non-collinear points, does there exist a line containing exactly two of the points?

With a given set of points, Voronoi diagram is a technique of partitioning a plane into set of points having the same closest sites. A number of studies are made on the generalization of Voronoi diagram. These studies are mainly conducted on different kind of distance metric. Some studies are also made upon Voronoi diagram of sites those are not necessarily points. The current researchers present a new generalization of the construction of Voronoi diagram in the plane with axes parallel rectangles. Here, the interest has been shifted from points to simple convex orthogonal blocks i.e. rectangles. The current researchers have computed the partition of the set of the rectangles in the plane into sets of lines having the same closest sites (where sites are rectangles in the plane. This paper represents a method to find the Voronoi diagram, which considered rectangles as sites. Prototype implementation has been done and the results are quite encouraging. The algorithm is easy to implement and finds application in VLSI layouts.

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all non-tangent lines to C through an external point to C, the set of all non-tangent lines to C through a point in C, and, for square q, the set of all non-tangent lines to C belonging to a Baer subplane PG(2,√q) with √q+1 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmaros [9] characterizing the pencil of lines through an internal point to C as the unique set of lines, up to projectivities, which provides a partition of the set of all non-internal points to C. However, the proof is not analogous, since it does not rely on the famous Lemma of Tangents of Segre which was the main ingredient in [9]. The main tools in the present paper are certain partitions in conics of the set of all internal points to C, together with some recent combinatorial characterizations of blocking sets of non-secant lines, see [2], and of blocking sets of external lines, see [1].

The notion of a centerpoint of a finite set of points in two and higher dimensions is a generalisation of the concept of the median of a (finite) set of points on the real line. In this paper, we present an algorithm for computing a centerpoint of a set of n points in the plane. The algorithm has complexity O(n) which significantly improves the O(n log3 n) complexity of the previously best known algorithm. We use suitable modifications of the ham-sandwich-cut algorithm and the prune-and-search technique to achieve this improvement.

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

On the boundary of a finite set of points in the plane

Determination of minimum number of sensors and their locations for an automated facility: An algorithmic approach

If $$\mathbb{X}$$ is a finite set of points in a multiprojective space $$\mathbb{P}^{n_1 } \times \cdots \times \mathbb{P}^{n_r } $$ withr ≥ 2, then $$\mathbb{X}$$ may or may not be arithmetically CohenMacaulay (ACM). For sets of points in ℙ1 × ℙ1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space.We show that each classification for ACM points in ℙ1 × ℙ1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM

Computing the shape of a planar points set

Chapter 20 - Voronoi Diagram Distances

Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

Given a finite set of points in R3, polyhedronization deals with constructing a simple polyhedron such that the vertices of the polyhedron are precisely the given points. In this paper, we present randomized approximation algorithms for minimal volume polyhedronization (MINVP) and maximal volume polyhedronization (MAXVP) of three dimensional point sets in general position. Both, MINVP and MAXVP, problems have been shown to be NP-hard and to the best of our knowledge, no practical algorithms exist to solve these problems. It has been shown that for any point set S in R3, there always exists a tetrahedralizable polyhedronization of S. We exploit this fact to develop a greedy heuristic for MINVP and MAXVP constructions. Further, we present an empirical analysis on the quality of the approximation results of some well defined point sets. The algorithms have been validated by comparing the results with the optimal results generated by an exhaustive searching (brute force) method for MINVP and MAXVP for some well chosen point sets of smaller sizes. Finally, potential applications of minimum and maximum volume polyhedra in 4D printing and surface lofting, respectively, have been discussed.

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearest- and farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-sided convex polygon, and study the nearest- and farthest-site Voronoi diagrams for sets of line segments and convex polygons. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n disjoint line segments is O(nm), which is asymptotically equal to that of the Voronoi diagram of n point sites with respect to the same distance function. In addition, we generalize this result to the Voronoi diagram of disjoint convex polygonal sites. We show that the combinatorial complexity of the nearest-site Voronoi diagram of n convex polygonal sites, each having at most k sides, is \(O(n(m+k))\). Finally, we show that the corresponding farthest-site Voronoi diagram is a tree-like structure with the same combinatorial complexity.

We study the minimum diameter problem for a set of inexact points. By inexact, we mean that the precise location of the points is not known. Instead, the location of each point is restricted to a continuous region (\({Imprecise}\) model) or a finite set of points (\({Indecisive}\) model). Given a set of inexact points in one of \({Imprecise}\) or \({Indecisive}\) models, we wish to provide a lower-bound on the diameter of the real points.

The discrete multi-agent rendezvous problem we consider in this paper is concerned with a specified set of points in the plane, called dwell-points, and a set of mobile autonomous agents with limited sensing range. Each agent is initially positioned at some dwell-point, and is able to determine its distances to dwell-points within sensing range, and also the bearings of those dwell-points with respect to a local reference of the agent. An agent can also determine if each dwell-point within sensing range is occupied by at least one agent. We do not, however, assume that an agent can count the number of agents occupying a dwell-point. We say the agents have rendezvoused if all the agents have moved to a set of dwell-points all within sensing radius of each other. The goal is to devise distributed motion strategies for each agent which will cause all the agents to rendezvous in the sense just defined, without any active communication among the agents. We propose a distributed motion rule and use graphs to characterize a class of dwell-points in the plane for which rendezvous is guaranteed under the rule.